Select the expression which is not equivalent to: \[ a^{2} b^{-3} c^{1.5} \] a) $\frac{a^{1.5} c^{2}}{\sqrt[4]{a^{-2} b^{12} c^{2}}}$ b) $\frac{\left(a^{-2} b^{-2} c^{3}\right)\left(a^{-2} b^{2} \sqrt{c^{-3}}\right)}{a^{-6} b^{2}}$ c) $\frac{a^{3} b^{-1} c^{3}}{\sqrt{a^{2} b^{4} c^{3}}}$
Solution
Step 1 ：First, we need to simplify each option to see if they are equivalent to \(a^{2} b^{-3} c^{1.5}\).
Step 2 ：For option a), we have: \(\frac{a^{1.5} c^{2}}{\sqrt[4]{a^{-2} b^{12} c^{2}}} = a^{1.5 - (-0.5)} b^{0 - 3} c^{2 - 0.5} = a^{2} b^{-3} c^{1.5}\). So, option a) is equivalent to the given expression.
Step 3 ：For option b), we have: \(\frac{\left(a^{-2} b^{-2} c^{3}\right)\left(a^{-2} b^{2} \sqrt{c^{-3}}\right)}{a^{-6} b^{2}} = a^{-2 - (-2) - 6} b^{-2 - 2 + 2} c^{3 - 1.5} = a^{2} b^{-3} c^{1.5}\). So, option b) is also equivalent to the given expression.
Step 4 ：For option c), we have: \(\frac{a^{3} b^{-1} c^{3}}{\sqrt{a^{2} b^{4} c^{3}}} = a^{3 - 1} b^{-1 - 2} c^{3 - 1.5} = a^{2} b^{-3} c^{1.5}\). So, option c) is also equivalent to the given expression.
Step 5 ：Since all options are equivalent to the given expression, there is no option that is not equivalent to \(a^{2} b^{-3} c^{1.5}\).
